162 I)RS. GUY BARLOW AND IT. B. KEENE ON THE ANALYSIS OF SOUND. 
Now take n near \n 1 and put 3 n — n x — p. Tlien 
y' = ~ sin n x t + — {cos (2 n—p) t— cos (4 n—p) £ + g- cosy>£ — ^ cos (6n—p) t 
A IT 
+ g cos (2 n+p) t. — } cos (8 n—p) t+A cos (4n+p) t — &c.}. 
Hence the frequencies are 
p, n l , 2 n + p, 4 n + p, Qn + p, &c. 
T'lie galvanometer responds to p ; this is the third order subharmonic of amplitude 
A137i. The beating pairs all beat with the same frequency 2 p, but their amplitudes 
are more unequal than in the case of the fundamental response. 
(3) Harmonic Series. 
Let the original current be 
y = Aj sin np A 2 sin 2 nj -j- A 3 sin 3 n^t + &c. 
without regard to phase differences between the constituents. 
The interrupted current is now made up of groups of components, there being one 
group associated with each harmonic. It is sufficient to consider only the case of 
synchronism with the fundamental of the series, i.e. when n — n x nearly. Putting 
n — n Y — p as before, the resulting tones may be tabulated thus :■—- 
Amp. ratio. 
A r 
A 2 . 
A s . 
a 4 . 
K 
1 
%■ 
p 2 n—p 
2 n v 
n—2p 3n—2p 
3 n v 
2n—3p 4n—3 p 
4%. 
3 n —4 p bn —4 p 
5 n v 
4w—5 p bn—bp 
* 
2 n-\-p 
4 n—p 
n-\-2p bn—2p 
3 p 6n—3p 
n—ip In. —4j> 
2n—bp 3n—bp 
1 
5 
4 n+p 
6 n—p 
3n-\-2p In —2 p 
2n-\~3pt 8n— 3p 
n-\-ip 9 n —4 p 
bp 1 On—bp 
1 
6 n-\-p 
Sn—p 
bn-\-2p 9n—2p 
in-\-3p 10 n —3 p 
3w+4p lln—ip 
2n-\-bp \2n—bp 
The interrupted current may be regarded as made up of 
(а) \ (A x sin n x t + A 2 sin 2 n x t -|- &c.). 
This is the original note with half the amplitude. 
(б) - (Ai cos pt + lA-i cos 3 pt -j- ;iA 5 cos 5pt + &c.). 
71 
This is the galvanometer response, which now has a complex character due to the 
superimposed subharmonics of 3n,, 5n u &c. 
